Semi-analytical Finite Element Method Research Articles

Research on detection of switch rail defects based on pulse reflection method of the dominant mode

Guided wave structural health monitoring offers a promising way for continuous surveillance and identification of structural damage. However, due to the variable cross-section properties of the switch rail in the longitudinal direction, there are different modes present on different cross-sections, so the guided wave signals excited are very complex. The traditional baseline subtraction method can be used to extract defect echo features, but it is prone to false positives due to the influence of external factors such as field temperature change and environmental noise. This paper proposes a switch rail defect detection method based on the pulse reflection method of the dominant mode. Initially, eight typical cross-sections are chosen based on the cross-sectional dimensions of the switch rail. Semi-analytical finite element(SAFE) method is utilized to solve the dispersion curves and wave structure matrix of guided waves in the switch rail. According to the 3σ rule, the dominant mode whose node amplitude is much higher than other modes is selected, and the excitation signal is applied at the maximum value of the mode, which successfully excites the dominant mode. Through simulation and experimental validation, the reflected waves of the dominant mode at the defect are observable and visible, and the pulse reflection method can be used to detect the defects at the switch rail foot with a positioning error of less than 10 cm. The pulse reflection method of the dominant mode proposed in this paper, which has high detection efficiency and is not affected by the ambient temperature, provides a new idea and way to detect defects of the switch rail based on the ultrasonic guided wave method.

Detection of edge delamination in composite laminates using edge waves

Detecting near-edge damage in composite structural elements using guided wave-based techniques can be challenging, primarily due to the complexity of wave analysis arising from material anisotropy. Furthermore, scattered waves containing damage information can be contaminated by waves reflected from the edges, which makes detecting near-edge damage difficult to implement. In the literature, studies showed that in elastic materials, the edge of a structure can serve as a waveguide, enabling the existence of typical edge modes with concentrated energy at the edge. However, studies regarding edge waves in composite structures have received limited attention. This paper aims to explore the potential of detecting edge delamination damage in composite laminates using edge waves. The modal properties of edge waves in [(0/90)2]s composite laminates are investigated using the Semi-Analytical Finite Element (SAFE) method. Additionally, dispersion curves for quasi-isotropic composite laminates are calculated. Following this, numerical and experimental studies were conducted to investigate the sensitivity of edge waves in detecting edge delamination in the [(0/90)2]s composite laminates. The outcomes of this study offer physical insights into the modal properties of edge waves and confirm their effectiveness in detecting damage near the edges.

In Situ Non-Destructive Stiffness Assessment of Fiber Reinforced Composite Plates Using Ultrasonic Guided Waves.

The multimodal and dispersive character of ultrasonic guided waves (UGW) offers the potential for non-destructive evaluation of fiber-reinforced composite (FRC) materials. In this study, a methodology for in situ stiffness assessment of FRCs using UGWs is introduced. The proposed methodology involves a comparison between measured wave speeds of the fundamental symmetric and antisymmetric guided wave modes with a pre-established dataset of UGW speeds and translation of them to corresponding stiffness properties, i.e., ABD-components, in an inverse manner. The dispersion relations of guided waves have been calculated using the semi-analytical finite element method. First, the performance of the proposed methodology has been assessed numerically. It has been demonstrated that each of the independent ABD-components of the considered laminate can be approximated with an error lower than 10.4% compared to its actual value. The extensional and bending stiffness properties can be approximated within an average error of 3.6% and 9.0%, respectively. Secondly, the performance of the proposed methodology has been assessed experimentally. This experimental assessment has been performed on a glass fiber-reinforced composite plate and the results were compared to mechanical tensile and four-point bending tests on coupons cut from the plate. Larger differences between the estimated ABD-components according to UGW and mechanical testing were observed. These differences were partly attributed to the variation in material properties across the test plate and the averaging of properties over the measurement area.

Study on monitoring broken rails of heavy haul railway based on ultrasonic guided wave.

Real-time monitoring of broken rails in heavy haul railways is crucial for ensuring the safe operation of railway lines. U78CrV steel is a common material used for heavy haul line rails in China. In this study, the semi-analytical finite element (SAFE) method is employed to calculate the dispersion curves and modal shapes of ultrasonic guided waves in U78CrV heavy steel rails. Guided wave modes that are suitable for detecting rail cracks across the entire cross-section are selected based on the total energy of each mode and the vibration energy in the rail head, web, and foot. The excitation method for the chosen mode is determined by analyzing the energy distribution of the mode shape on the rail surface. The ultrasonic guided wave (UGW) signal in the rail is analyzed using ANSYS three-dimensional finite element simulation. The group velocity of the primary mode in the guided wave signal is obtained through continuous wavelet transform to confirm the existence of the selected mode. It is validated that the selected mode can be excited by examining the similarity between the vibration shapes of a specific rail section and all modal vibration shapes obtained through SAFE. Through simulation and field verification, the guided wave mode selected and excited in this study demonstrates good sensitivity to cracks at the rail head, web, and foot, and it can propagate over distances exceeding 1 km in the rail. By detecting the reflected signal of the selected mode or the degree of attenuation of the transmitted wave, long-distance monitoring of broken rails in heavy-haul railway tracks can be achieved.

Damage assessment in L-shaped bends by nonlinear feature guided waves

L-shaped bends are crucial lightweight structural components in aerospace manufacturing, renowned for their high specific strength and stiffness. However, the geometric configuration of these bends makes them susceptible to stress concentration in the bending zone. Effective technique for damage assessment of the bending region of L-shaped bends is highly demanded. This paper proposes a nondestructive testing method using nonlinear ultrasonic feature guided waves (FGWs) for early damage assessment of L-shaped bends. Semi-analytical finite element (SAFE) method and perfectly matched layer (PML) technique are employed to analyze FGW propagation in L-shaped bends. Three-dimensional finite element models are utilized to investigate the FGWs propagation in the bending region of L-shaped bends in a numerical perspective. Cumulative second harmonic generation (SHG) of FGWs is successfully observed by selecting suitable mode pairs based on synchronism and non-zero power flux conditions, which provides a promising alternative for early damage assessment in L-shaped bends. Experimental findings are consistent with the numerical ones. Subsequently, thermal damage and plastic deformation with varying gradients were introduced into the bending region of L-shaped bends. The use of SHG of selected FGW mode has effectively assessed the damage with an improved sensitivity, compared to conventional linear acoustic one. This investigation substantiates the potential of nonlinear FGWs in detecting early damage in L-shaped bends, offering promise for future industrial testing applications.

A semi-analytical wavelet finite element method for wave propagation in rectangular rods

Prior knowledge of the dispersion curves and mode shapes of guided waves provides valuable information for wave mode selection and excitation in the field of non-destructive evaluation (NDE) and structural health monitoring (SHM). They are typically computed by the matrix methods, the finite element (FE) and semi-analytical finite element (SAFE) methods. However, the former is prone to numerical instability, and the latter two are limited by the refinement level of the FE mesh. In this paper, a semi-analytical wavelet finite element (SAWFE) method is presented to characterize wave propagation in rectangular rods. The piecewise polynomial interpolation functions of the SAFE method are replaced by two-dimensional scaling functions of the B-spline wavelet on the interval (BSWI). To demonstrate the accuracy of the proposed SAWFE technique, the propagation of guided waves in an aluminium plate is studied first. Then, the propagation of guided waves in rectangular rods of arbitrary aspect ratio is investigated. The results of this work clearly show that the SAWFE method presented here has higher accuracy and efficiency than the SAFE method.

Propagation of the Fundamental Lamb Modes in Strain Stiffened Hard-Magnetic Soft Plates

Abstract This study focuses on investigating hard-magnetic soft materials, characterized by magnetoactive polymers containing magnetically polarized particles as fillers. The research utilizes the Gent model of hyperelasticity to analyze the propagation of Lamb waves in a magnetically induced deformed compressible plate. In this investigation, we explore both finite deformations and incremental wave propagation in nonlinear hard-magnetic soft materials. The main objective is to formulate the elastic tensor and relevant wave equations within the framework of Lagrangian space. To assess the dispersion characteristics of the guided wave, the study introduces and discusses an extension of the semi-analytical finite element (SAFE) method. Using this numerical approach, the research further examines the effects of magnetic flux densities and its orientation with respect to wave propagation direction on the dispersion characteristics of the fundamental Lamb modes. The study starts by examining the limiting case of the neo-Hookean material model to explain such inherent dependencies. These dependencies are then further emphasized by including the strain-stiffening effect that the Gent material model describes. The research findings reveal the presence of a threshold applied magnetic flux, beyond which the Gent-type material may undergo a snap-through instability, resulting in changes in the dispersion characteristics of the fundamental symmetric Lamb mode.

Hydrodynamic forces in higher modes of a thin cantilever beam resonator

The hydrodynamic force in a thin uniform beam cantilever is essential quantity for designing a resonator. We obtained the hydrodynamic drag forces in a uniform cantilever vibrating at the first four transverse modes using a semi-analytical boundary element method (BEM) and finite element method (FEM) in ANSYS. The present three-dimensional (3D) BEM inertial and damping forces are compared with FEM over a frequency range with less than 9.5% deviation until frequency parameter of 100; thereafter, it deviates numerically due to neglecting the non-linear convective forces. Considering the hydrodynamic damping force, the quality factor is estimated for the first four transverse modes of a uniform cantilever beam. As a result, the present 3D BEM model behaves well at the first bending mode compared to FEM. We also study the effect of the aspect ratio (ratio of width to length) of the beam for all four modes. However, the first mode quality factor sufficiently correlates with the numerical value for all aspect ratios. For uniform beams operating with higher modes, the influence of nonlinear and resonance frequency by the numerical model is more pronounced at smaller lengths. Hence, the 3D BEM approach can estimate the hydrodynamic forces on cantilever-based resonators, for example, atomic force microscopy and biosensors.

Propagation characteristics of circumferential feature guided waves propagation in a revolved topological waveguide

In our previous research, propagation characteristics and combined harmonic generation of feature guided waves (FGWs) in welded joints of plate structure, were analyzed theoretically and observed numerically. Circumferential ultrasonic guided waves find extensive applications in assessing rotating structures, highlighting the importance of investigating the propagation characteristics of circumferential feature guided waves (CFGWs) in such revolved topological waveguides. In this study, propagation characteristics of CFGW in revolved topological waveguides are investigated. Semi-analytical finite element (SAFE) method in conjunction with a perfectly matched layer (PML) within a cylindrical coordinate system is used to explore the modal and dispersion characteristics of CFGWs. Second-harmonic generation (SHG) of CFGWs is further analyzed and discussed both in theoretical and numerical manner. Cumulative SHG of selected CFGW mode pairs, propagation in the welded joint of the revolved topological waveguide, is observed successfully. The obtained results demonstrate that the energy trapping effect of CFGWs in the welded joint of revolved topological waveguide amplifies the generation of cumulative second harmonics, thereby enhancing the measurability of SHG of CFGW. This investigation reveals the potential of damage assessment in revolved topological waveguides by CFGWs.

Determination of Young’s Modulus of PET Sheets from Lamb Wave Velocity Measurement

BackgroundThe elastic modulus of polyethylene terephthalate (PET) sheets is typically measured through destructive tests that require specific sample preparation and time-consuming testing procedures.ObjectiveTo improve the efficiency of measuring the elastic modulus of PET sheets, research on a non-destructive measurement approach using guided Lamb waves was conducted.MethodsIn this approach, the group velocity of the zero-order symmetric Lamb wave mode (S0 mode) at a single frequency is first measured from PET sheets. The semi-analytical finite element method (SAFEM) is used as the forward model to calculate the corresponding numerical group velocity. Particle swarm optimisation (PSO) is used to update the elastic modulus in the SAFEM model until the numerical group velocity from the model matches the experimental results.ResultsThe results show that measuring the group velocity data at a single frequency is sufficient for elastic modulus measurement while the material thickness can be assumed as a constant, which improves the efficiency of the measurement. The identified modulus differs from the tensile modulus of the material due to the frequency dependence of the elastic modulus. However, this discrepancy could be eliminated by using a linear regression model.ConclusionsThe method mentioned above can achieve non-destructive and efficient measurement of the elastic modulus of PET sheets, which can potentially be applied for in-line quality inspection in PET bottle production processes.

Guided wave-based crack detection in U-shaped flexural plate butt welds

Based on the guided wave, a crack detection method for U-shaped flexural plate (UFP) butt welds is established to realize damage localization and quantitative identification. Firstly, the semi-analytical finite element (SAFE) method was used to obtain the dispersion curve in a plate with weld by discretizing the cross-section and then solving the characteristic equation. Subsequently, numerical studies were performed to investigate the mode conversion and energy transfer efficiency in weld bending. Then the method was determined after the wave packet analysis of voltage responses under different damage conditions. Finally, a validation experiment was conducted in an actual UFP butt weld. The result indicates that there exist 10 guided wave modes in a plate with weld at 150 kHz, while only 2 modes in a flat plate; For curvature radius of weld bending substantially greater than the weld thickness, the guided wave energy mainly passes through the bending in the form of transmitted waves and the reflected wave formed in the bending can be ignored; The damage can be located with the defect echo of the specific guided wave and the error distance only accounting for 1.5% of the total weld length. It is effective to take wavelet energy as an indicator to characterize the damage degree. The method significantly improves crack detection efficiency and provides a scientific basis for monitoring similar structures.

Exciting and Detecting Higher-Order Guided Lamb Wave Modes in High-Density Polyethylene Structures Using Ultrasonic Methods.

High-density polyethylene (HDPE) pipes are becoming increasingly popular, being used in various fields, such as construction, marine, petroleum, water transfer, process water, methane gas collection, oil and gas gathering, gas distribution systems, mining, acid and wet gas lines, offshore oil and gas and in nuclear power plants. Higher-order guided Lamb wave (UGW) modes can be used to detect various defects in complex structures. We will apply this methodology to one of the types of plastic-the structure of high-density polyethylene (HDPE). However, the excitation of UGW modes faces numerous challenges, especially when there is a need to identify which mode is excited. It is essential to note that, in the higher frequency range, multiple different higher-order modes can usually be excited. This can make it difficult to determine which modes have actually been excited. The objective of this research was to successfully excite and receive various higher-order UGW modes in high-density polyethylene structures using both ultrasonic single-element transducers and a phased array. Theoretical calculations were performed using a variety of methods: semi-analytical finite element (SAFE) method, 2D spatial-temporal spectrum analysis and finite element modeling (FEM). The results obtained from both measurements and simulations clearly demonstrate the possibility of efficiently exciting and receiving different Lamb wave modes possessing different phase velocities.

Prediction of critical strains of flexible pavement from traffic speed deflectometer measurements

Structure indicators of flexible pavements can be categorized into material moduli, deflections, distresses, or responses. Since field measurements of pavement responses, such as strains and stresses, require embedded sensors, predictions from easily measured surface deflections are desired. This study aims to predict critical strains of flexible pavements from traffic speed deflectometer (TSD) measurements. Semi-analytical finite element method (SAFEM) was developed and validated for simulation of truck-pavement system by comparison with traditional three-dimensional (3-D) FEM. Results indicated that SAFEM saved significant calculation time while ensuring the same accuracy. The efficient advantage makes it possible to develop an extensive databased of TSD measurement considering a wide rangte of pavement layer thickness, material moduli, temperature, and speed. Traditional regression analysis and artificial neural networks (ANN) models were applied to predict critical tensile, shear, and compressive strains from surface deflection parameters. The developed models were further validated with field measurements. The regression models with explicit forms are easier to understand but require asphalt layer thickness as an input for prediction of tensile or shear strains. Nevertheless, the ANN model offers greater accuracy and only requires the inputs of TSD-measured deflection slopes, which is more applicable for pavement management applications.

SOLUTION OF PHYSICALLY NONLINEAR PROBLEMS OF DEFORMATION OF MASSIVE AND THIN-WALLED PRISMATIC BODIES

The results of a numerical study on test examples of the convergence of the developed version of the semi-analytical method of finite elementsare presented. A comparison of machine time and accuracy of solving elastic and plastic problems obtained on the basis of finite elements with variableand averaged mechanical parameters was performed.The convergence of the finite element method and the semianalytical finite element method in the calculation of prismatic bodies loaded with localized and distributed external influences was studied. An effective numerical approach to the study of arbitrarily loaded massive and thin-walled prismatic bodies of complex shape, the deformation of which can pass beyond theelastic limit of the material, is being developed based on the finite element moment scheme and the semianalytical variant of the finite element method. Due to the representation of displacements by polynomials and the use of iterative methods of solving systems of equations, this approach is developed in relation to the calculation of objects with arbitrary boundary conditions on the ends, which made it possible to expand the area of effective application of the semi-analytical method of finite elements to a new class ofproblems.A number of new complex problems of elastic and elastic-plastic deformation of massive and thinwalled prismatic bodies, which have an independentapplied value, have been solved.

Quantitative assessment of corrosion-induced wall thinning in L-shaped bends using ultrasonic feature guided waves

The utilization of L-shaped bends has become widespread due to their versatility and ease of application in various industrial fields. Nevertheless, the geometric configuration of these bends makes them susceptible to stress concentration in the bending zone, which can lead to wall thinning caused by accelerated corrosion. This poses a profound challenge to the bearing capacity and reliability of L-shaped bends and highlights the urgent necessity for an effective technique to assess corrosion-induced wall thinning in the bending zone. In this study, we propose a nondestructive testing (NDT) method based on ultrasonic feature guided waves (FGWs) for the evaluation of local corrosion-induced wall thinning in L-shaped bends. Propagation characteristics of FGWs in L-shaped bends were analyzed using the semi-analytical finite element (SAFE) method and perfectly matched layer (PML) technique. Time-reversal (TR) method was employed to address the dispersion and multi-modes phenomenon of the selected guided wave modes and improve the signal-to-noise ratio (SNR) of measured FGW signals. Enhanced SNR of two types of FGW modes signals at different frequencies were verified by experimental and numerical results. Obtained results reveal that FGWs can effectively detect corrosion-induced wall thinning inside L-shaped bends, with improving the signal reflection coefficient by TR method.

Determination of internal efforts in the base finite elements of SAFEM

The principles of calculating the internal efforts of a circular finite element in the semi-analytical finite element method (FEM) based on the obtained components of the stress tensor and the peculiarities of the approach associated with the use of the moment scheme of the finite element (FEM) are considered. Formulas for determining longitudinal, shear forces, bending and torque moments have been obtained.
 A special place, among the variety of objects considered with the help of analytical and numerical methods, is occupied by bodies of revolution of complex shape and cross-sectional structure, formed by the movement of some creative surface along a closed or opened line without breaks. The selected geometric class is used as natural structures of nodes and details in construction of mechanical engineering. The sufficiently wide distribution of the specified forms in the construction and machine-building industries, on the one hand, and the possibility of significantly simplifying the solving relationships by taking into account their geometric features, on the other hand, provide a basis for the development and use of various modifications of the finite element method (FEM). The semi-analytical finite element method (SAFEM) is one such approach that has gained widespread use for solving problems whose objects are prismatic bodies and bodies of revolution of complex shape and cross-sectional structure. Due to the introduction of additional hypotheses that do not reduce the accuracy of the approximation, the representation of deformations and stresses in physical terms and in accordance with the moment scheme of the finite element (MSFE), on the one hand, it is possible to avoid the time-consuming procedure of numerical integration over the cross-sectional area of the finite element (FE), on the other hand - maintain the high efficiency of 3D discretization.
 Despite the large number of publications devoted to the semi-analytical method of finite elements, the question of determining internal forces, which are often component factors of the strength criteria laid down in state building codes, is inappropriately neglected. The use of SAFEM in combination with МSFЕ creates some mathematical features of calculating internal longitudinal, shearing forces and moments.

Analysis of structures with arbitrary kinematic boundary conditions by the semi-analytical finite element method

The successful application of FEM to the analysis of structures is largely due to the efficiency of the use of modern software packages, in connection with which the role of program systems that implement the solution process increases. The correct organization of the computing complex, the choice of optimal algorithms for solving systems of linear and nonlinear equations largely determine the possibilities of the method in terms of the structural complexity of the objects under consideration, the accuracy of the results obtained, and the complexity of setting nonlinear problems. Therefore, there is an increased interest in the development of fairly universal computing complexes based on FEM. One of the effective software complexes is the "Strength" system, designed to conduct comprehensive research in the field of mechanics of a deformable solid, the basic principles of construction of which are used in this work in the implementation of a semi-analytical version of the FEM.
 In this work, solutions of a significant number of control problems of deformation of massive and thin-walled prismatic bodies under different boundary conditions and loads are obtained. In the process of solving new problems, the estimation of the convergence of results was carried out on the basis of a sequential increase in finite elements and contained terms of decomposition, an increase in the accuracy of systems of linear and nonlinear equations, and the accuracy of satisfaction with natural boundary conditions was checked. The developed effective method for solving new complex problems of deformation of prismatic bodies is implemented in the form of complex programs and can be used in design and construction practice in construction, mechanical engineering and other fields of technology.

Implementation and analysis of viscoelastic damping in a 2D + 1D model of railway track vibrations

One of the most important technological challenges in the railway context nowadays is reducing the noise produced by the interaction between the train wheels and the track surface, that is the rolling noise. Introducing viscoelastic damping in the railway structure is one method to mitigate this rolling noise. In the paper, we investigate the impact of damping in the vibrational response of railway tracks using different viscoelastic models and parameters available in the literature. For this purpose, the main focus here is on developing an advanced numerical model in which viscoelastic models representing the rail foundation and/or pad are implemented in the framework of the Semi-Analytical Finite Element (SAFE) method. Linear or nonlinear viscoelastic models are considered, compared and effectively implemented for accelerance computation. The obtained numerical strategy leads to affordable and accurate predictions on dynamic features of the railway structure compared to classical methods.

Investigation of energy trapping effect for nonlinear guided waves in a topographical structure

Cumulative second harmonic generation (SHG), where the amplitude of the generated second harmonic increases with propagation distance, is commonly utilized to enhance the measurability of nonlinear guided waves by providing a sufficient signal-to-noise ratio. Investigating and confirming the cumulative SHG of feature guided waves (FGWs) in topographical waveguides is crucial for accurately assessing material nonlinearity in such structures. This study examines the cumulative SHG of FGWs propagating in a typical topographic structure of a welded joint through both numerical simulations and experimental investigations. A theoretical model is developed using the semi-analytical finite-element (SAFE) method and perfectly matched layer (PML) to analyze the propagation characteristics of FGWs in a welded joint. The numerical and experimental results provide verification of the “trapping effect” of FGWs within the welded joint. Furthermore, the “cumulative effect” of SHG of FGWs is observed. Remarkably, both the experimental and numerical findings demonstrate that cumulative SHG of FGWs can be achieved by employing suitable mode selection and frequency tuning in the topographic structure, which aligns well with the theoretical predictions. This investigation contributes to advancing our understanding of cumulative SHG in FGWs propagated within topographical structures. These findings have practical implications for accurately characterizing material nonlinearity in topographic waveguides.

Flood impact on structural response of asphalt pavement: A finite element modeling approach

The frequency and intensity of floods have increased due to climate change. These natural events negatively affect asphalt pavements, submerging their layers and reducing structural resistance. This work introduces a computational finite element model suitable for investigating the impact of floods on asphalt pavement response. Unlike previous works, which consider a multi-layer analysis, in this research, the resilient modulus varies locally as a function of moisture, which could be variable in the transverse direction of the cross-section. The moisture distribution in the pavement structure is found by solving the Richards’ equation through non-linear finite element analysis. This moisture distribution is coupled to the three-dimensional elasticity problem to find the structural response to traffic loads. The structural problem was solved using the semi-analytical finite element method. Results show that the reduction of the resilient modulus in the unbound and subgrade layers is heterogeneous, with the areas near the edges of the pavement being the most affected. It was also found that the increase in elastic surface deflection depends on both the load position and time, which suggests the damage caused by vehicles could be different for each wheel path, depending on the time in which the vehicles circulate. This research presents a valuable tool for evaluating the impact of flooding on asphalt pavement structures.